Problem - 2841
Let $P(x)$ be a polynomial with integer coefficients satisfying that both $P(0)$ and $P(1)$ are odd. Show that $P(x)$ has no integer zeros.
If this is not true, then there exists an integer $k$ such that $P(k)=0$.
If $k$ is even, then $P(k)\equiv P(0)\equiv 1\pmod{2}$. And, if $k$ is odd, then $P(x)\equiv P(1)\equiv 1\pmod {2}$. Therefore, regardlessly, it always hold that $$P(k)\equiv 1\pmod{2}$$
This means $P(k)\ne 0$, or $k$ is not a zero of $P(x)$.